Nth Term of GP

Q. If three distinct numbers $a,b,c$ are in G.P. and the equations $a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$ have a common root, then which one of the following statements is correct ?

1. $d,e,f$ are in G.P.
2. $d,e,f$ are in A.P.
3. $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in G.P.
4. $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in A.P.

Q. The nth term of a GP 18, -12, 8, ...... is $\frac{512}{729}$. The value of n is .

1. 7
2. 8
3. 9
4. 10

Q. Three positive numbers form an increasing GP. If the middle term of the GP is doubled, then new numbers are in AP. Then, the common ratio of the GP is ___.

1. $\sqrt{2}+\sqrt{3}$
2. $3+\sqrt{2}$
3. $2-\sqrt{3}$
4. $2+\sqrt{3}$

Q. Let $t_n$ represents the $n^{th}$ term of a G.P. If $t_3=2$ and $t_6=-\frac{1}{4},$ then the value of $t_{10}$ is

1. $-\frac{1}{128}$
2. $-\frac{1}{64}$
3. $\frac{1}{128}$
4. $\frac{1}{64}$

Q. Which of the following biconditional statements are true?

1. $2\ne 3\leftrightarrow 2>3\mathrm{or}2<3$
2. $2=3\leftrightarrow 3=4$
3. $a<3\leftrightarrow a>3$
4. $a=3\leftrightarrow a\le 3\mathrm{and}a\ge 3$

Q. Let $S_1,S_2,...S_n$ be squares such that for each $n\ge 1$, the length of a side of $S_n$ equals the length of the diagonal of $S_{n+1}$. If the length of a side of $S_1$ is $10$ cm, then the least value of $n$ for which the area of $S_n$ less that 1 sq cm

1. $7$
2. $8$
3. $9$
4. $10$

Q.

If the $5^{th}$ term of a G.P. containing positive terms is $\frac{1}{3}$ and $9^{th}$ term is $\frac{16}{243}$, then the $4^{th}$ term will be

1. $\frac{3}{4}$

2. $\frac{1}{2}$

3. $\frac{1}{3}$

4. $\frac{2}{5}$

Q. In an increasing geometric series, the sum of the second and the sixth term is $\frac{25}{2}$ and the product of the third and fifth term is $25.$ Then, the sum of $4^{\text{th}},6^{\text{th}}$ and $8^{\text{th}}$ terms is equal to :

1. $35$
2. $30$
3. $26$
4. $32$

Q.

If $(p+q{)}^{\text{th}}$ term and $(p-q{)}^{\text{th}}$ term of a G.P. be $m$ and $n,$ then the $p^{\text{th}}$ term will be ___.

1. $\frac{m}{n}$

2. $\sqrt{mn}$

3. mn

4. o

Q. The 6th term from the end of the sequence 4, 8, 16, 32, ....., 16384 is .

1. 1024
2. 512
3. 256
4. 64

Q. In an increasing geometric progression, the sum of the first term and the last term is $66$, the product of the second terms from the beginning and the end is $128$ and sum of all terms is $126$. Then the number of terms in the progression is

1. $5$
2. $6$
3. $7$
4. $8$

Q. If three distinct numbers $a,b,c$ are in G.P. and the equations $a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$ have a common root, then which one of the following statements is correct ?

1. $d,e,f$ are in G.P.
2. $d,e,f$ are in A.P.
3. $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in G.P.
4. $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in A.P.